Angular Size

and related topics

If we use a magnifying glass with a relaxed eye the image is at
infinity. How then can we determine the magnification produced by this lens? What is most
important to us as users of the magnifying glass is the size of the final image on our
retina. To explain a bit more about this we will also need to introduce the concept of
"angular magnification". Figure 6a shows an object being imaged by an eye. If we
consider the eye as merely a lens at the eye-air interface, the linear magnification of
the object is l '/ l as expected. What matters to us
most is the linear size of the image on the retina. This, as can be seen in the diagram,
is determined by the angle subtended by the object. A 1 cm object at 1 m produces the same
size of final image on the retina as a 10 cm object at 10 m. This is why the "angular
size" q of an object viewed by eye is often
an important quantity.

Imaging by an eyeball. The angle subtended by the object determines the
linear size of the final image on the retina.

How can a magnifying glass help? The largest angle that the object can
subtend without a magnifying glass is q = h/D, where h
is the height of the object, and D is the near point of distinct vision, as you
can fill in below.

Left: The largest image that the eye can get unaided, here the object
is at the near point of distinct vision, and the angular size of the object as seen by the
eye is h/D
Right: the image produced with a relaxed eye and a magnifying glass. Now the angular
size of the image is given by h / f'

If we can use the magnifying glass to increase the apparent value of
the angular size of the object, we are on to a winner (though in fairness the concept has
been around well over two thousand years). Let us consider placing our eye right up to a
magnifying glass that has been set up as per the diagram above. In this case the rays
reaching our eye come from an image at infinity, so the eye can be relaxed. The angular
size of the image is h/f '. The angular magnification is then the ratio
of the size of this angle to the biggest angle that could be obtained by the unaided eye
ie

q

/q
= (h/f ) / (h/D) = D / f .

Thus if the focal length of the magnifying glass is less than your
near-point, you can produce an image of the object on your that is larger with the
magnifying glass than without, at the same time as being able to have a relaxed eye rather
than a strained one.

If you are prepared to accommodate your eye to look at an image
appearing at your near point of distinct vision, the final image on your retina will be
bigger still. Verify yourself that the angular magnification caused by the
magnifying glass in this case is 1 + D/f .